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{"name": "Integration: via mechanics and area", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: via mechanics and area", "tags": [], "metadata": {"description": "<p>Q1. True/false questions about basic facts.</p>\n<p>Q2 and Q3. Velocity-time graphs are given with areas underneath them shaded. The area of the shaded regions are given. From this, definite integrals of v ar eto be determined.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>See Lectures 13.3 and Lecture 14.3.</p>\n<p></p>\n<p>The key&nbsp;facts needed to do these questions are:</p>\n<p>-The area between the graph and the $t$-axis corresponds to <em>distances</em>.</p>\n<p>-If the velocity is positive, then it means the particle is moving forwards, i.e., that the change in position is positive.</p>\n<p>-If the velocity is negative, then the particle is moving backwards, i.e., the change in position is negative.</p>", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"ar22": {"name": "ar22", "group": "Ungrouped variables", "definition": "siground((1/3*a2*(x22+2)^3 + c2*(x22+2))-(1/3*a2*x22^3 + c2*x22),3)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(-4..-1)", "description": "", "templateType": "anything"}, "x41": {"name": "x41", "group": "Ungrouped variables", "definition": "a4", "description": "", "templateType": "anything"}, "x42": {"name": "x42", "group": "Ungrouped variables", "definition": "b4", "description": "", "templateType": "anything"}, "ar41": {"name": "ar41", "group": "Ungrouped variables", "definition": "siground(0.5*(  ( 1/4*(x42)^4 - 1/3*(a4+b4+c4)*(x42)^3 + 1/2*(a4*b4+a4*c4+b4*c4)*(x42)^2 - a4*b4*c4*(x42) )  -  ( 1/4*(x41)^4 - 1/3*(a4+b4+c4)*(x41)^3 + 1/2*(a4*b4+a4*c4+b4*c4)*(x41)^2 - a4*b4*c4*(x41) )  ),3)", "description": "", "templateType": "anything"}, "ar32": {"name": "ar32", "group": "Ungrouped variables", "definition": "siground(0.5*(  (1/3*(x32+2)^3 - 1/2*(a3+b3)*(x32+2)^2 + a3*b3*(x32+2) )  -  (1/3*(x32)^3 - 1/2*(a3+b3)*(x32)^2 + a3*b3*(x32))  ),3)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "ar42": {"name": "ar42", "group": "Ungrouped variables", "definition": "siground(0.5*(  ( 1/4*(x42)^4 - 1/3*(a4+b4+c4)*(x42)^3 + 1/2*(a4*b4+a4*c4+b4*c4)*(x42)^2 - a4*b4*c4*(x42) )  -  ( 1/4*(x42+2)^4 - 1/3*(a4+b4+c4)*(x42+2)^3 + 1/2*(a4*b4+a4*c4+b4*c4)*(x42+2)^2 - a4*b4*c4*(x42+2) )  ),3)", "description": "", "templateType": "anything"}, "x32": {"name": "x32", "group": "Ungrouped variables", "definition": "b3", "description": "", "templateType": "anything"}, "x11": {"name": "x11", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything"}, "x31": {"name": "x31", "group": "Ungrouped variables", "definition": "b3-random(2..3)", "description": "", "templateType": "anything"}, "x22": {"name": "x22", "group": "Ungrouped variables", "definition": "random(1..4 except 2)+x21", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "x21": {"name": "x21", "group": "Ungrouped variables", "definition": "random(-4..-2)", "description": "", "templateType": "anything"}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "a4+random(1..3)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(-3..-1)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "x12": {"name": "x12", "group": "Ungrouped variables", "definition": "random(2..5) + x11", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "ar21": {"name": "ar21", "group": "Ungrouped variables", "definition": "siground((1/3*a2*x22^3 + c2*x22)-(1/3*a2*x21^3 + c2*x21),3)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "a3+random(3..4)", "description": "", "templateType": "anything"}, "ar1": {"name": "ar1", "group": "Ungrouped variables", "definition": "(0.5*a1*x12^2 + b1*x12) - (0.5*a1*x11^2 + b1*x11)", "description": "", "templateType": "anything"}, "ar31": {"name": "ar31", "group": "Ungrouped variables", "definition": "siground(-0.5*(  (1/3*(x32)^3 - 1/2*(a3+b3)*(x32)^2 + a3*b3*(x32) )  -  (1/3*(x31)^3 - 1/2*(a3+b3)*(x31)^2 + a3*b3*(x31))  ),3)", "description": "", "templateType": "anything"}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "b4+2", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "b1", "x11", "x12", "ar1", "a2", "c2", "x22", "x21", "ar21", "ar22", "a3", "b3", "x31", "x32", "ar31", "ar32", "a4", "b4", "c4", "x41", "x42", "ar41", "ar42"], "variable_groups": [], "functions": {"plotgraph": {"parameters": [["q", "number"], ["x1", "number"], ["x2", "number"], ["ymin", "number"], ["ymax", "number"], ["a", "number"], ["b", "number"], ["c", "number"]], "type": "html", "language": "javascript", "definition": "// Shading under a graph! This functions plots a graph of y = a(x-r1)(x-r2)\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar xmin = -7;\nvar xmax = 7;\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n  '600px',\n  '500px',\n {\n   boundingBox: [xmin,ymax,xmax,ymin],\n   axis: false,\n   showNavigation: false,\n   grid: true\n }\n);\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\n/*var xaxis = brd.create('axis',[[0,0],[1,0]],\n                       {name:'t', withLabel: true, label: {position:'rt', offset: [-10, 5]}}\n\t\t\t);*/\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\n/*var yaxis = brd.create('axis',[[0,0],[0,1]],\n                      {name:'v', withLabel: true, label: {offset: [-10, 220]}}\n                      );*/\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n\n\n// This function shades in the area below the graph of f\n// between the x values x1 and x2\n\nvar shade = function(f,x1,x2,colour) {\n  var dataX1 = [x1,x1];\n  var dataY1 = [0,f(x1)];\n\n  var dataX2 = [];\n  var dataY2 = [];\n  for (var i = x1; i <= x2; i = i+0.1) {\n      dataX2.push(i);\n      dataY2.push(f(i));\n  }\n\n  var dataX3 = [x2,x2];\n  var dataY3 = [f(x2),0];\n\n  dataX = dataX1.concat(dataX2).concat(dataX3);\n  dataY = dataY1.concat(dataY2).concat(dataY3);\n\nvar shading =  board.create('curve', [dataX,dataY],{strokeWidth:0, fillColor:colour, fillOpacity:0.2});\n\nreturn shading;\n}\n\n\n//Define your functions\nvar f1 = function(x) {\n  return a*x+b;\n}\n\nvar f2 = function(x) {\n  return a*x*x + c;\n}\n\nvar f3 = function(x) {\n  return 0.5*(x-a)*(x-b);\n}\n\nvar f4 = function(x) {\n  return 0.5*(x-a)*(x-b)*(x-c);\n}\n\n\n//Plot the graph and do shading\nswitch(q) {\n    case 1:\n        board.create('functiongraph', [f1]);\n        shade(f1,x1,x2, 'red');\n        break;\n    case 2:\n        board.create('functiongraph', [f2]);\n        shade(f2,x1,x2,'red');\n        shade(f2,x2,x2+2,'green');\n        break;\n    case 3:\n        board.create('functiongraph', [f3]);\n        shade(f3,x1,x2,'red');\n        shade(f3,x2,x2+2,'green');\n        break;\n    case 4:\n        board.create('functiongraph', [f4]);\n        shade(f4,x1,x2,'red');\n        shade(f4,x2,x2+2,'green');\n        break\n}\n\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Select those options which are correct/true. $v(t)$ is the velocity of some object.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "1", "minAnswers": "4", "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "choices": ["In a velocity-time graph, areas are equal to distance travelled", "To calculate change in position based on area, you need to think about whether $v$ is positive or negative for the different areas.", "$\\displaystyle \\int_{0}^{10} v(t) dt$ is the change in position from $t=0$ to $t=10$.&nbsp;", "To get from velocity to position, you have to integrate.", "In a position-time graph, areas are equal to distance travelled", "In a position-time graph,&nbsp;gradients are equal to speeds", "$\\displaystyle \\int_{0}^{10} v(t) dt$ is the distance travelled from $t=0$ to $t=10$.&nbsp;", "To get from velocity to position, you differentiate"], "matrix": ["1", "1", "1", "1", "-4", "-4", "-4", "-4"], "distractors": ["", "", "", "", "", "", "", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>{plotgraph(2,x21,x22,-5,25,a2,0,c2)}</p>\n<p></p>\n<p>This is a velocity-time graph of a particle.&nbsp;</p>\n<p>The area of the&nbsp;left shaded region is $\\var{ar21}$.&nbsp;The area of the right shaded region is $\\var{ar22}$.</p>\n<p>What is $\\displaystyle \\int_{\\var{x21}}^{\\var{x22}} v(t) dt$? [[0]]</p>\n<p>What is $\\displaystyle \\int_{\\var{x21}}^{\\var{x22+2}} v(t) dt$?[[1]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{ar21}", "maxValue": "{ar21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{ar22+ar21}", "maxValue": "{ar22+ar21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>{plotgraph(3,x31,x32,-3,7,a3,b3,0)}</p>\n<p></p>\n<p>This is a velocity-time graph of a particle.&nbsp;</p>\n<p>The area of the&nbsp;left shaded region is $\\var{ar31}$. The area of the right shaded region is $\\var{ar32}$.</p>\n<p>What is the total <em></em>distance&nbsp;travelled by the particle between&nbsp;$t=\\var{x31}$ and&nbsp;$t=\\var{x32+2}$? [[0]]</p>\n<p>What is $\\displaystyle \\int_{\\var{x31}}^{\\var{x32}} v(t) dt$?&nbsp;[[1]]</p>\n<p>What is $\\displaystyle \\int_{\\var{x31}}^{\\var{x32+2}} v(t) dt$? [[2]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{ar31+ar32}", "maxValue": "{ar31+ar32}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{-ar31}", "maxValue": "{-ar31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{ar32-ar31}", "maxValue": "{ar32-ar31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}]}